Works both ways. There are publishing functions and development functions.
A "studio" does development. It can be independent or owned by a bigger company.
A "publisher" focuses on handling distribution / sales & marketing functions. It can publish games from studios it owns ("first party") or from external studios it has business contracts with ("third party").
There are companies that do various mixes of these functions, but the biggest tend to do them all.
In addition, I think in the last few years the swing has been towards bigger publishers or corporate holding groups (Embracer, Microsoft/Sony, Tencent, etc) sweeping up lots of previously independant studios (Eidos, Riot, Double Fine, Mojang, etc) in acquisitions - for the sake of owning the intellectual property and freedom to do more with the license. But also, and crucially, stopping other rival companies doing the same thing first, leading to big arguments like this one: https://www.theverge.com/2023/3/3/23623363/microsoft-sony-ft...
Perhaps the colder global financial situation will cool this acquisition/centralisation spree, but maybe not!